A local characterization of simply-laced crystals

Abstract

We provide a simple list of axioms that characterize the crystal graphs of integrable highest weight modules for simply-laced quantum Kac-Moody algebras. 0. Introduction Following Kashiwara [K2], a crystal is an edge-colored directed graph satisfying a simple set of axioms. Every (integrable highest weight) representation of a sym-metrizable quantum Kac-Moody algebra has a crystal associated to it that encodes important combinatorial data. For example, knowing the crystal of a representation allows one to immediately deduce tensor product and branching rules involving that representation. There are a number of explicit constructions known for the crystals of representations of particular quantum algebras; e.g., [KN], [L3]. In the cases of finite type (i.e., quantum enveloping algebras of semisimple Lie algebras), it is also possible to give explicit descriptions of crystals in terms of the integer points of a convex polytope [BZ], [L4]. In the general case, Littelmann's Path Model [L1], [L2] provides an algorithm to generate the crystal of any representation. On the other hand, the crystals of representations form a very special subclass of the set of all crystals, and it has been an open problem to find a simple set of local axioms that characterize them. In other words, can one determine from local structural conditions whether a crystal graph is the crystal of a representation? In this paper, we give an affirmative solution in the simply-laced cases; i.e., for quantum Kac-Moody algebras with a Cartan matrix whose off-diagonal entries are 0 or −1. These simply-laced crystals are arguably the most important, since all highest weight crystals of finite or affine type-the ones of widest interest-are either simply-laced or may be obtained from such crystals by a standard technique of "folding" by diagram automorphisms. Among the properties we use to characterize these crystals, (P1)-(P3) are equivalent to the defining axioms for a general crystal (aside from the fact that we have not explicitly required the assignment of weight vectors to the vertices). The key additional axioms are two relations (P5)-(P6) and their duals (P5)-(P6) that may be viewed as combinatorial analogues of the Serre relations. Roughly speaking , they require that for each (distinct) pair of raising operators E i and E j , one